Skew-Hermitian matrix
In , a with entries is said to be skew-Hermitian or antihermitian if its is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relation where A^{\text{H}} denotes the conjugate transpose of the matrix A . In component form, this means that |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} for all indices i and j , where a_{ij} is the element in the j -th row and i -th column of A , and the overline denotes . Skew-Hermitian matrices can be understood as the complex versions of real , or as the matrix analogue of the purely imaginary numbers. The set of all skew-Hermitian n \times n matrices forms the u(n) , which corresponds to the Lie group . The concept can be generalized to include s of any with a . Note that the of an operator depends on the considered on the n dimensional complex or real space K^n . If (\cdot|\cdot) denotes the scalar product on K^n , then saying A is skew-adjoint means that for all u,v \in K^n one has (Au|v) = - (u|Av) \, . s can be thought of as skew-adjoint (since they are like 1 \times 1 matrices), whereas s correspond to operators. Example For example, the following matrix is skew-Hermitian : A = \begin{bmatrix} -i & 2 + i \\ -2 + i & 0 \end{bmatrix} because : -A = \begin{bmatrix} i & -2 - i \\ 2 - i & 0 \end{bmatrix} = \begin{bmatrix} \overline{-i} & \overline{-2 + i} \\ \overline{2 + i} & \overline{0} \end{bmatrix} = \begin{bmatrix} \overline{-i} & \overline{2 + i} \\ \overline{-2 + i} & \overline{0} \end{bmatrix}^\mathsf{T} = A^\mathsf{H} Properties * The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are . Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal. * All entries on the of a skew-Hermitian matrix have to be pure ; i.e., on the imaginary axis (the number zero is also considered purely imaginary). * If A and B are skew-Hermitian, then is skew-Hermitian for all a and b . * A is skew-Hermitian if and only if i A (or equivalently, -i A ) is . * A is skew-Hermitian if and only if the real part \Re{(A)} is and the imaginary part \Im{(A)} is . * If A is skew-Hermitian, then A^k is Hermitian if k is an even integer and skew-Hermitian if k is an odd integer. * A is skew-Hermitian if and only if x^\mathsf{H} A y = -y^\mathsf{H} A x for all vectors x,y . * If A is skew-Hermitian, then the e^A is . * The space of skew-Hermitian matrices forms the u(n) of the U(n) . Decomposition into Hermitian and skew-Hermitian * The sum of a square matrix and its conjugate transpose \left(A + A^\mathsf{H}\right) is Hermitian. * The difference of a square matrix and its conjugate transpose \left(A - A^\mathsf{H}\right) is skew-Hermitian. This implies that the of two Hermitian matrices is skew-Hermitian. * An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B : :: C = A + B \quad\mbox{with}\quad A = \frac{1}{2}\left(C + C^\mathsf{H}\right) \quad\mbox{and}\quad B = \frac{1}{2}\left(C - C^\mathsf{H}\right) References Category:Advanced mathematics